Continuity of the $L_p$ Balls and an Application to Input-Output System Described by the Urysohn Type Integral Operator

Abstract

In this paper the continuity of the set valued map $p\to B_{\Omega ,\mathcal{X},p}(r),$ $p\in (1,+\infty ),$ is proved where $B_{\Omega ,\mathcal{X},p}(r)$ is the closed ball of the space $L_p\left(\Omega ,\Sigma ,\mu ;\mathcal{X}\right)$ centered at the origin with radius $r,$ $\left(\Omega ,\Sigma ,\mu \right)$ is a finite and positive measure space, $\mathcal{X}$ is separable Banach space. An application to input-output system described by Urysohn type integral operator is discussed.